Re: What is the interval between ℕ and ω when doubled?

Le 14/04/2024 à 21:25, Richard Damon a écrit :

On 4/14/24 2:08 PM, WM wrote:

Le 13/04/2024 à 16:48, Mikko a écrit :

On 2024-04-13 12:13:07 +0000, WM said:
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\omega is not an element of |N.

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That is true. The question concerns the distance between both.
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The second set does not contain \omega.

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But it contains ω*2.

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What size has the interval from sweet to blue?

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Are they points on the ordinal axis?

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No, sweet, blue, and ℕ are not points on the ordinal axis.

 But ω and all elements of ℕ are points on the ordinal axis.

 ω only exist on that TRANSFINITE ordinal axis, not the finite ordinal axis.

Some ordinal numbers of the beginning of the sequence (with k, m, n  ) are:
0, 1, 2, 3, ..., ,  + 1, ...,  + k, ...,  +  (= 2), 2 + 1, ..., k, ..., k + m, ...,  (= 2), 2 + 1, ..., 2 + , ..., 2 + k + m, ..., 22, ..., 2k + m + n, ..., 3 + 2k + m + n, ..., k, .., ,  + 1, ..., k, ..., +1, +1 + 1, .., k, ..., 2, ..., , ...,  (= 0), 0 + 1, ..., 00, ..., 000, ..., 000 (= 1), 1 + 1, ..., 111 (= 2), ..., 1, ... .
Better readable in Transfinity, https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p.42.
Regards, WM